Problem 53 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$\sqrt{x^{2}+4 x+4}$$ (Hint: Factor the polynomial first.)

Step-by-Step Solution

Answer

|x + 2|

Step 1: Understand the Expression

We need to simplify the expression \( \sqrt{x^2 + 4x + 4} \). The hint suggests we should factor the polynomial under the square root first.

Step 2: Factor the Quadratic Polynomial

The expression under the square root is \( x^2 + 4x + 4 \). We will factor this quadratic polynomial. The factors of \(4\) (the constant term) that add up to \(4\) (the linear coefficient) are \(2\) and \(2\). So, the factored form is \( (x + 2)(x + 2) \) or \((x + 2)^2\).

Step 3: Substitute the Factored Form

Replace the quadratic expression with its factored form in the original expression: \( \sqrt{x^2 + 4x + 4} = \sqrt{(x + 2)^2} \).

Step 4: Simplify the Square Root

The square root of a square is the absolute value of the base: \( \sqrt{(x + 2)^2} = |x + 2| \). This is because square roots yield non-negative results.

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into its simplest components, often a product of simpler polynomials. Let’s explore how to factor quadratic polynomials, which are polynomials of the form \(ax^2+bx+c\). Here, the goal is to express the quadratic as a product of two binomials.

• Look for two numbers that multiply to the constant term \(c\) and add to the linear coefficient \(b\).
• In the expression \(x^2 + 4x + 4\), you need numbers that multiply to \(4\) and add to \(4\). Both numbers are \(2\).
• This leads to the factors \((x + 2)(x + 2)\), often written as \((x + 2)^2\).

Factoring helps in simplifying expressions and is particularly useful when dealing with equations, as it allows for easy solving by setting factors to zero.

Square Roots

Square roots simplify expressions by finding a value that, when multiplied by itself, gives the original number or expression. In our exercise, we started with the square root of the factored polynomial: \( \sqrt{(x + 2)^2} \).

• The square root and the square are inverse operations. Therefore, \( \sqrt{(x + 2)^2} \) simplifies to \(|x + 2|\).
• The absolute value is used because square roots must yield non-negative results, ensuring that \(|x + 2|\) remains positive regardless of whether \(x + 2\) is positive or negative.

Understanding square roots simplifies many algebraic expressions, allowing us to work with expressions in their cleanest form.

Quadratic Expressions

Quadratic expressions are integral in algebra and often take the form \(ax^2 + bx + c\). They appear frequently in real-world problems and mathematical modeling.

• A key feature of quadratics is that they can generate parabolic graphs, helping model various phenomena.
• Factoring quadratic expressions, such as \(x^2 + 4x + 4\) into \((x + 2)^2\), often makes solving equations more manageable.
• Simplified quadratic expressions are easier to evaluate, differentiate, and integrate, especially when dealing with calculus applications.

Overall, mastering the manipulation and simplification of quadratic expressions through techniques like factoring is foundational in both algebra and advanced mathematics.